MOTION OF A PARTICLE 287 Substituting in equation (VI) from equations (1) and (2) we get F=_8o^m. (3) Therefore the force varies inversely as the fifth power of the distance from the center of force. The negative sign in the second member of equation (3) shows that the force is directed towards the origin; in other words, it is an attractive force. PROBLEMS. 1. Show that if a particle describes the reciprocal spiral rd = a in a central field of force, the force is attractive and varies inversely as the cube of the distance from the origin, which is the center of attraction. 2. Show that if a particle describes the logarithmic spiral r = ea9 in a central field of force, the expression for the force is F = — ——^~—-• 3. A particle moves in a central field of force where the force is away from the center and is proportional to the distance. Show that the orbit is a hyperbola. 4. Show that in the preceding problem -the radius vector sweeps over equal areas in equal intervals of time. 5. A particle describes an ellipse in a field of force the center of which is at the center of the ellipse. Show that the force varies directly as the distance and is directed towards the center. 6. In the preceding problem show that the radius vector sweeps over equal areas in equal intervals of time. 7. A particle describes an ellipse in a field of force, the center of which is at one focus. Show that the force is towards the center of force, and is inversely proportional to the square of the distance. 220. Motion of Two Gravitating Particles. — Suppose two particles of masses m and M to move under the action of their mutual gravitational attraction, as in the case of the sun and the earth or the earth and the moon. Then if r is the distance between the centers and y the gravitational constant the mutual force of attraction is mM In order to fix our ideas let M be the mass of the sun and m the mass of the earth. Then the sun gives the earth